TEACHING MA THEMA TICS WITH THE BRAIN IN MIND: LEARNING PURE MA THEMA TICS WITH MEANING AND

UNDERSTANDING

TANYA JOHNSON

B. Ed., University of New Brunswick, 1993 DAUS, University of New Brunswick, 1994

A Project Submitted to the School of Graduate Studies

of the University of Lethbridge in Partial Fulfillment of the

Requirements for the Degree

MASTER OF EDUCATION

F ACUL TY OF EDUCATION

LETHBRIDGE,ALBERTA

January 2003

I dedicate this work to

My Mom

When I thought I did not have the time and energy to finish my project; the strength and

perseverance shown by my mom with her battle with cancer became my inspiration to

keep going and to never give up in life.

111

Abstract

This culminating master's project applies data and information discovered in a content

analysis of research documents to create a brain-based pure math teacher resource that

will help teachers teach the pure mathematics 20 program with meaning and

understanding. The resource includes a rationale, as well as explanations for the brain-

based mathematics lesson framework. Teacher friendly daily lessons are laid out in a

thematic unit for the algebraic equations, relations and functions section of the

curriculum. The resource utilizes a brain-based approach to teaching and learning

providing teachers with an easy to understand, practical, everyday guide that can easily

be implemented into the pure math classroom. This resource is needed because students

continue to feel inadequate and inferior in pure math classrooms across Alberta. Changes

needed to resolve this disturbing situation include teachers themselves altering their

teaching strategies to help minimize the existing problems in the pure math program, and

this project contributes to the knowledge about improving best teaching practices.

Research on the multiple intelligence theory reminds us of the different student learning

styles and the fact that, more than one type of teaching strategy should be used to deliver

the pure math program. Current research on the science of learning has brought to light

some very interesting ideas of how a student's brain works and the applications of this

work to classroom practice. As teachers, we can translate this information into classroom

practice in order to help our students learn pure mathematics with meaning and

understanding.

IV

Acknow ledgements

To my husband, Kevin, thank you for your financial and emotional support. As

well, I am grateful for the time you took out of your own busy schedule to drive with me

to Lethbridge over the past few years.

Thank you to my principal, Bonnie Annicchiarico, for her gift of time. You

always provided the extra time I needed in order to meet deadlines. It was very much

appreciated.

Thank you to my supervisors, Leah Fowler and Craig Loewen, for your guidance

and academic support.

v

Table of Contents

Dedication ....................................... " ........................................... " ...... iii

Abstract .............................................................................................. .i v

Acknowledgements .................................................................................. v

Table of Contents .................................................................................... vi

Chapter One: The Beginnings ................................ " ............. '" ..................... 1

Introduction ............ '" ............. " .................................................... 1

Background: The Journey Begins ......................................................... 1

The Alberta Mathematics Program .............................................. 1

The Differences: Pure and Applied Mathematics ............................... 2

Rationale: Why Brain-Based Teaching? ................................................. 2

Pure Mathematics: A Disturbing Situation ..................................... 2

Mathematical Anxiety: Lasting Effects ......................................... 3

Another Approach: Brain-Based Teaching .................................... ..4

Starting Point: Brain-Based Teaching ......................................... .4

Caution: Teachers Need To Analyze The Brain-Based Approach .......... 5

Teaching Styles Parallel Learning Styles: Mathematics for all Students .... " ...... 6

One Teaching Style: Who Loses? ................................................ 6

Varied Teaching Styles: Who Wins? ............................................ 7

Teaching Mathematics with the Brain in Mind .......................................... 8

Personal Reflections ............................................................... 8

Project Purpose ..................................................................... 8

Words: Contextual Meaning ...................................................... 9

VI

Research Question ................................................................ 11

Chapter Summary .......................................................................... 11

Chapter 2: Literature Review ..................................................................... 12

Introduction ................................................................................. 12

Why Brain-Based Teaching: Different Learners ...................................... 12

Student's Learning Styles ....................................................... 13

The Multiple Intelligence Theory .............................................. 13

Brain-Based Learning ............................................................ 14

Overlaps: Learning Styles, MI, Brain-Based Learning ..................... 15

Learning Pure Mathematics with Meaning and Understanding ...................... 15

Facilitating Learning: Brain-Based Teaching .......................................... 17

Applying Brain Research to Classroom Practice ............................. 18

How Our Brains Learn ............................................................ 19

The Importance of Teachers Using Previous Learned Knowledge ......... 20

Brain Research: Attention and Learning ....................................... 21

Brain Research: Emotion and Learning ....................................... 22

Brain Research: Movement and Learning .................................... 23

Brain Research: Rehearsal and Learning ...................................... 24

Brain Research: Memory and Learning ....................................... 25

Brain Research: Elaboration and Learning .................................... 25

Brain Research: Elaborative Rehearsal Strategies ........................... 26

Brain-Based Teaching and Learning Models .......................................... 27

Researchers: Caine and Caine ................................................... 27

Vll

Researcher: Jensen ............................................................... 28

Researcher: Erlauer ............................................................... 29

Researcher: Hardiman ............................................................ 29

Researcher: King-Friedrichs .................................................... 30

Chapter Summary ......................................................................... 31

Chapter 3: Brain-Based Teacher Resource ...................................................... 32

Introduction ................................................................................. 32

Teacher Resource: Underlying Learning Principles .................................. 32

Recognize Learning Styles ...................................................... 32

Learning Pure Mathematics with Meaning and Understanding ............ 33

Brain-Based Teaching and Learning .......................................... 33

Framework: Brain-Based Daily Lessons ............................................... 34

Brain-Based Strategy #1: Previous Learned Knowledge .................... 34

Brain-Based Strategy #2: Attention and Learning ........................... 34

Brain-Based Strategy #3: Emotion and Learning ............................ 35

Brain-Based Strategy #4: Movement and Learning .......................... 36

Brain-Based Strategy #5: Rehearsal and Learning ........................... 36

Brain-Based Strategy #6: Memory and Learning ............................ 37

Brain-Based Strategy #7: Elaboration and Learning ........................ 37

Brain-Based Strategy #8: Collaboration and Learning ...................... 38

Brain-Based Lesson Plans ................................................................. 38

Mathematics Project: Quadratic Equations and Functions .................. 39

Lesson One ........................................................................ 45

Vlll

Lesson Two ........................................................................ 48

Lesson Three ...................................................................... 53

Lesson Four ........................................................................ 58

Lesson Five ........................................................................ 62

Lesson Six ......................................................................... 66

Lesson Seven ..................................................................... 71

Summary .................................................................................... 76

References .................................................................................. 77

IX

Chapter One: The Beginnings

"Students must learn mathematics with understanding, actively building new knowledge

from experience and prior knowledge."

(National Council of Teachers of Mathematics, 2000, p. 20)

Introduction

Throughout chapter one, a solid foundation is created for the idea of teaching

mathematics with the brain in mind. Key background information sets the stage for the

compelling rationale for teaching pure mathematics with the brain in mind. Ideas

associated with brain-based teaching are discussed. The reasoning behind how teaching

styles should parallel learning styles is brought to the forefront in brain-based teaching so

all students have the opportunity to learn pure mathematics with meaning and

understanding. One tool in your toolbox approach to teaching will not meet the learning

needs of all students. Justification for the use of various teaching strategies is reinforced

throughout the chapter. The afore mentioned interrelated brain-based concepts give birth

to a pure mathematics teacher resource that will help teachers teach the pure mathematics

program with meaning and understanding.

Background: The Journey Begins

The Alberta Mathematics Program

The high school mathematics program has just undergone many curriculum

changes over the past five years. Alberta Learning implemented a new mathematics

program that included an applied mathematics stream and a pure mathematics stream.

The high school mathematics programs were developed with the advice and support of

many individuals and groups: teachers, school administrators, post-secondary educators,

1

parent groups, representatives of professional organizations, such as the Alberta

Teachers' Association, Mathematics Council Alberta Teachers' Association and the

Alberta Chamber of Resources, and other members of the community. The programs

were designed to help students develop skills to solve a wide range of problems both

inside and outside of mathematics courses, and to keep pace with the latest information

technology.

The Differences: Pure and Applied Mathematics

2

The main differences between applied mathematics and pure mathematics are the

topics studied, and the approach to solving problems and developing understanding. Both

programs help students develop the critical skill of using mathematics to find solutions to

problems in real-life situations. Students planning to pursue programs requiring the study

of mathematics at the post-secondary level need to be registered in the pure mathematics

stream. Pure mathematics is a requirement for post-secondary level programs that possess

a connection to mathematics. Pure mathematics courses emphasize the specialized

language of algebra as the preferred method for learning mathematical concepts and for

solving problems. Students learn about mathematical theories, find exact value solutions

to equations, and use formal mathematical reasoning and models in problem solving.

Therefore, all the time and money that was put into this new high school mathematics

program will ensure its survival for a long time to come.

Rationale: Why Brain-Based Teaching?

Pure Mathematics: A Disturbing Situation

During the last four years of teaching pure mathematics, the following

observations of students are a familiar teaching experience: blank stares, frightened

3

looks, frustrated body language, and streaks of tears. Does every pure mathematics

student encounter these symptoms of stress? Of course not, but each day I see students

suffering through taxing encounters with the subject. As a teacher, I am disturbed by

these student expressions of difficultly in learning. Therefore, I question the pure

mathematics program and my mathematical teaching approach. Obviously, some students

feel inadequate and inferior when subjected to the pure mathematics curriculum and I

want to discover teaching strategies that will enhance their learning.

Mathematical Anxiety: Lasting Effects

Based on my own classroom practice of teaching the pure mathematics program

over the last five years, students undergo a high level of mathematical anxiety each day.

Students are fearful of what they are learning in class. Fear paralyzes both learning and

teaching in the classroom. Caine & Caine (1990) claim recent research on how the brain

learns supports the importance of classroom environments; threat and stress can inhibit

learning. Students are being turned off mathematics. Some students report hating

mathematics. Evidently, these intense emotions have lasting effects, because students are

becoming complacent and satisfied with just passing grades. Some students are dropping

out of the pure mathematics program. Furthermore, student whose dreams involve

professions with a connection to mathematics are locking these imaginings into a black

box and pursuing something else. Producing life-long mathematics learners has not been

a realistic outcome of the program. Although, this scenario of failure does not occur in

every pure mathematics class, I argue it happens more often than teachers would like to

recognize. The situation is affecting the well-being and the future knowledge, careers,

and skills of our students. Obviously, something needs to change. What is the solution to

this problem? Should we change the curriculum or change our mathematical approach to

teaching? or both?

Another Approach: Brain-Based Teaching

4

The question needs to be asked: what can be done to change this unsettling pattern

of failure and dropout in pure mathematics? The same question of mathematical learning

difficulties has been constantly etched in my mind since I started teaching the pure

mathematics program. The solution to the question has been researched in current

educational journals and books, spoken about by experts in the mathematical field, and

experienced with fellow teachers. There are probably many solutions to enhancing

mathematics learning; ranging from trying to motivate students to work harder to

counselling more students to take the applied mathematics route. I am a teacher so I am

interested in what teachers can do to help minimize the problems that exist in the pure

mathematics program. One central solution lies in my research assumptions that teachers

can change the teaching strategies they use to deliver the pure mathematics program to

their students. Current research by E. Jensen (1995, 1997, 1998,2000), P. Wolfe (1998,

1999,2000,2001), R. Sylwester (1990, 1994, 1995), and many others on the science of

learning brought to light some very interesting ideas of how a student's brain works and

the applications of this work to classroom practice. It is hoped that teachers can translate

this information into classroom practice in order to enable our students to learn with

meaning and understanding.

Starting Point: Brain-Based Teaching

Since teaching is complex and contextual, it is very hard to identify directly

proportional correlated relationships between specific teaching practices and student

success. There is no recipe that lists a set of ingredients and methods for good teaching

for all teachers. Teaching is a highly personal act. Teaching is a process of human

interaction involving the teacher and his or her students in particular contexts. As Parker

Palmer (1998) remarks, "The Courage to Teach" is built on the premise that good

teaching cannot be reduced to technique; good teaching comes from the identity and

integrity of the teacher. Therefore, the problem situation with the pure mathematics

program is more complex than a solution of altering teaching strategies based on brain-

based research can support. Nevertheless, according to brain researchers, brain-based

teaching is a starting point that uses the way students learn to formulate and structure

daily classroom lessons. As well, teachers themselves can participate in this solution.

Teachers can directly help their students learn the pure mathematics curriculum with

understanding. Pure mathematics teachers themselves can learn and teach mathematics

with meaning.

5

Caution: Teachers Need to Analyze the Brain-Based Approach

Other researchers like Bruer (1998) believe current brain research has little to

offer educational practice. New results in neuroscience point to the brain's lifelong

capacity to reshape itself in response to experience. Educators' great challenge is

developing learning environments and practices to exploit the brain's lifelong plasticity.

Some researchers feel brain science should return to the back burner. More research

needs to be done on how our brains learn before we can apply this knowledge to learning

in the classrooms. Even Jensen (2000), who has written numerous accounts in support of

brain research in the classroom, warns although neuroscience has much to offer teaching

and learning, teachers must remain cautious about applying lab research to classrooms.

Teachers need to verify the brain-based approaches to learning with their own depth of

knowledge and experience about the way their students learn.

Teaching Styles Parallel Learning Styles: Mathematics for all Students

One Teaching Style: Who Loses?

How is the pure mathematics program currently delivered to students in the

classroom? Of course, there is no one set way, although through my own experience,

many teachers teach pure mathematics primarily from a direct approach. The direct

approach includes the presentation and development of processes or algorithms for

carrying out mathematical tasks. Craig Loewen (2001), a professor at the University of

Lethbridge, explains that a typical lesson plan in a direct approach to teaching might

involve the presentation of a process or algorithm followed by a series of teacher-led

examples, and then by the assignment of a set of questions that directly apply the

algorithm or process. Direct teaching encourages students to memorize what is being

taught. Also, the mathematical content is being presented as a series of unrelated

concepts. Since everything traditionally, was learned in isolation and no relationships

were established amongst the mathematical content, our brains will simply forget what

we have learned over a short period of time. Furthermore, in a class of thirty students

there is more than one type of learning style. For example, Erualer (2003) reiterates the

typical classroom usually includes 46% visual learners, 35% kinaesthetic learners, and

19% auditory learners. As well, Brandt (1990) contends because students are unique and

will respond differently to a variety of instructional methods we need to respect the

individual differences among us so all students are given an opportunity to learn in their

own way. Therefore, if there is more than one student learning style in a classroom then

6

should there not be more than one teaching style to ensure understanding by all the

students. If only one teaching style is used to teach the pure mathematics program many

students will lose out on learning with meaning and understanding.

Varied Teaching Styles: Who Wins?

In contrast, teaching pure mathematics with the brain in mind establishes rich

connections amongst the mathematical processes being taught to the student. It is like

gi ving each student a key to a door in their brain that will allow them to learn the pure

mathematics curriculum with understanding. As educators, we recognize these different

types of learning. Some of what we teach in pure mathematics requires students to

engage in rote rehearsal learning. Examples are basic algebraic processes such as

operations with polynomials, and solving equations. However, much of the pure

mathematics curriculum cannot be memorized and be fully understood. For example,

understanding an algorithm in pure mathematics does not include the method of rote

rehearsal. Wolfe (2001) suggests for these types of learning, elaborative rehearsal

strategies are much more effective. Elaborative rehearsal strategies encourage the learner

to elaborate on the information in a manner that increases understanding and retention of

that information. For example, some elaborative rehearsal strategies include real-life

problem solving, making connections to previous learning, developing concepts by

making connections to other ideas, visual representations and models, math projects,

writing activities, scientific applications, mnemonic patterns, and motivated active

learning. In many instances, these strategies make the information more meaningful and

relevant to the learner. Elaborative rehearsal strategies connect meaning and emotion to

what is being learned in order to enhance retention. If varied teaching styles are used to

7

teach the pure mathematics program students will win; more students will learn

mathematics with meaning and understanding. Both R. Brandt's (1990, 1993, 1994,

1997, 1999,2000, and M. D' Arcangelo's (1998, 2000, 2001) research support these

ideas.

Teaching Mathematics with the Brain in Mind

Personal Reflections

8

With eight years of teaching experience, I feel developing a learning environment

with the brain in mind can be a successful approach to learning for both math teachers

and students. When I reflect about what enabled me to learn mathematical concepts with

understanding; I begin to realize these various brain strategies are what made the ideas

meaningful for me. As I observe the way students learn, these brain related strategies

make perfect sense of why it is easier for some students to approach learning in this

manner. Researchers such as Jensen (1998) continually review recent research and

theory on the brain and balance this information with tips and techniques for using the

information in classrooms. In many instances, brain-compatible learning is explored with

the teacher and student in mind.

Project Purpose

The purpose of my project is to develop a practical brain-based resource for pure

mathematics 20 teachers. More specifically, I want to investigate teaching the high school

pure mathematics 20 program with the student's brain in mind. I will develop a teaching

unit that applies brain-based teaching methods that enable mathematics teachers to teach

students pure mathematics with meaning and understanding. The teaching unit will be a

pure mathematics resource for teachers with daily brain-based lesson plans that correlate

9

with the objectives of the pure math 20 curriculum. The thematic unit in the teacher

resource will entail the curriculum area of algebraic quadratic equations, relations and

functions. The reason this unit of the pure math 20 curriculum was chosen is because a

lot of the mathematical concepts in this section require abstract thinking about theoretical

ideas. Many students find this part of the pure mathematics program to be challenging

and frustrating as this area of curriculum is in the most urgent need of development. I

believe that using brain-based elaborative rehearsal strategies to teach these mathematical

concepts will enable students to learn mathematics with meaning and understanding. The

daily lessons will be developed around a framework that approaches learning with the

brain in mind.

Words: Contextual Meaning

The meaning of some key words in my purpose must be explained in terms of the

context of my project. What does it mean when my project refers to teaching pure

mathematics with the brain in mind? Teaching pure mathematics with the brain in mind is

a teaching practice that incorporates research on how the brain learns and implements

these brain-based strategies in daily classroom lessons to help students learn mathematics

with meaning and understanding. Caine and Caine (1995) cite that brain-based teaching

and learning uses a holistic approach stressing the importance of how the brain learns in

order to produce meaningful learning.

The brain-based approach to learning incorporates elaborative rehearsal

strategies. Wolfe (2001) describes how these strategies encourage learners to elaborate on

the information being learned in a manner that increases understanding and retention of

that information. In my project, these strategies make the information more meaningful

and relevant to the learner. For example some elaborative rehearsal strategies involve

making meaning using associations, analogies, metaphors, and visual representations.

10

Using these teaching strategies, students begin to learn mathematics with meaning

and understanding. Loewen (2000) uses The National Council of Teachers of

Mathematics (NCTM) to describe understanding mathematics in terms of connections.

Connections can be made between mathematical ideas and other ideas, models, words,

algorithms, and the world. Our learning of a concept is based on the connections we

make cognitively with that concept and other related ideas. In this project, the more

connections students make the more they understand about what they have learned.

Furthermore, Hiebert and Carpenter (1992) define understanding as the way information

is represented and structured. Historically, and still today researchers believe that a

mathematical idea is understood if it is part of an internal network. Of course, the degree

of understanding is linked to the number and strength of these connections.

Learning mathematics with meaning deals with students making their own

individual meaning from a concept they are learning. The brain-based approach to

learning mathematics with meaning uses strategies to help students find their own

constructed meaning in a mathematical concept. Sousa (2001) states learning with

meaning is a result of how students relate content to their past knowledge and

experiences. Students break concepts down, compare concepts, and refer to concepts in a

different way until they make sense or are meaningful to them.

11

Research Question

What elaborative rehearsal strategies are needed to develop a practical resource of

daily lessons for pure math 20 teachers that includes a brain-based approach to teaching

and learning that enables students to learn pure math with meaning and understanding?

Chapter Summary

The high school mathematics program has just undergone many curriculum

changes over the past five years. Alberta Learning implemented a new mathematics

program that included an applied mathematics stream and a pure mathematics stream.

Based on my own classroom practice, students undergo a high level of mathematical

anxiety each day in the pure mathematical stream. Students are fearful of what they are

learning in class. I feel this is a disturbing situation. In order to minimize the problems

that exist in the pure mathematics program teachers can change the teaching strategies

they use to teach mathematics. Brain-based teaching is a starting point that uses the way

students learn to formulate and structure daily classroom lessons. Nevertheless, teachers

need to verify the brain-based approaches to learning with their own depth of knowledge

and experience about the way their students learn. Many teachers teach pure mathematics

primarily from a direct approach. In contrast, teaching pure mathematics with the brain in

mind establishes rich connections amongst the mathematical processes being taught to

the student. All of the aforementioned information will help to produce a brain-based

pure mathematics teacher resource that will allow teachers to teach the pure mathematics

program with meaning and understanding.

Chapter 2: Literature Review

"Students draw on knowledge from a wide variety of mathematical topics, sometimes

approaching the same problem from different mathematical perspectives or representing

the mathematics in different ways until they find methods that enable them to make

progress. "

(National Council of Teachers of Mathematics, 2000, p. 3)

Introduction

At the beginning of chapter two, an overview of educational researchers' thoughts

on students' learning styles, multiple intelligence theory, brain-based learning, and

learning pure mathematics with meaning and understanding are presented. The next

section investigates what researchers think about facilitating learning through brain-based

teaching. Researchers argue about the application of brain research to classroom practice.

As well, ideas are explored on topics such as: how our brains learn; importance of

teachers using previous learned knowledge; attention and learning; emotion and learning;

movement and learning; rehearsal and learning; memory and learning; and elaboration

and learning. Elaborative rehearsal strategies are offered for use in brain-based

classrooms. Lastly, researchers such as Caine and Caine (1990, 1995, 1997, 1998),

Jensen (1998), Erlauer (2003), Hardiman (2001), and King-Friedrichs (2001) explain

their brain-based teaching and learning models.

Why Brain-Based Teaching: Different Learners

Based on five years of classroom teaching in pure mathematics, I believe students

learn in different ways. Students possess their own unique ways of learning. Therefore, a

pure mathematics-learning environment should be structured so each student's learning

12

13

style is present. Imagine a mathematics classroom where all students are given an equal

opportunity to learn. Learning mathematics with the brain in mind recognizes different

student learning styles by presenting mathematical concepts through the varying manners

in which the brain learns.

Students' Learning Styles

Learning styles is a popular topic in educational research. Many researchers

believe students' learning styles should be an important consideration for daily lesson

planning. Brandt (1990) wrote an article on learning styles after conversing with Pat

Guild, senior author of ASCD's Marching to Different Drummers, and he advised

teachers to identify key elements of the student's learning style and match your

instruction and materials to those individual differences. As well, Brandt (1990)

concludes students are different, students will respond differently to a variety of

instructional methods, and we need to respect and honour the individual differences in

our classrooms. According to Shields (1993) teachers catering to particular learning

styles is just good use of common sense in the classroom. Teachers should validate

students' personal approaches to learning by varying their instruction. The approach is

high-quality lesson planning, you should reach all children through a variety of activities.

The Multiple Intelligence Theory

The multiple intelligence theory (MI) put forward by Gardner has been praised as

one of the most important ideas to help with school improvement. After a conversation

with Gardner, Checkley (1997) reviews what is involved in the multiple intelligences:

linguistic intelligence, logical-mathematical intelligence, spatial intelligence, bodily

kinaesthetic intelligence, musical intelligence, interpersonal intelligence, intrapersonal

14

intelligence, and naturalist intelligence. Sweet (1998) believes allowing students to use

their knowledge about how they learn best can increase their enthusiasm, raise their

achievement levels, and foster growth in their other intelligences. Silver, Strong, Perini

(1997) integrate Gardener's multiple intelligence theory and learning styles in order to

better understand how students learn. Learning styles are concerned with the process of

learning while the multiple intelligence theory looks at the content and products of

learning. The multiple intelligence theory (MI) acknowledges the differences in the way

each student's brain learns. MI supports teachers who use several teaching methods to

teach one concept so all students have the opportunity to learn that concept.

Brain-Based Learning

Kalbfleisch and Tomlinson (1998) remind educators of three principles from brain

research: emotional safety, appropriate challenge, and self-constructed meaning suggest

that a one-size-fits-all approach to classroom teaching is ineffective for most students.

Classrooms need to be responsive to students' varying readiness levels, varying interests,

and varying learning profiles. Furthermore, Kalbfleisch and Tomlinson (1998) explain

how teachers have a responsibility to present information in varied ways, for example,

orally, visually, through demonstration, part to whole, and whole to part if they want to

give all their students an opportunity to learn. Moreover, Given (2002) feels teachers can

stimulate and facilitate learning in all children by addressing the need to know in multiple

ways. By providing alternate ways of learning, students are at a liberty to gain new

information through those learning styles most comfortable for them.

15

Overlaps: Learning Styles, MI, Brain-Based Learning

Multiple intelligences, learning styles, and brain-based education are theories with

many overlaps. Chock-Eng and Guild (1998) state the application of the three theories

have similar outcomes. They judge multiple intelligences, learning styles, and brain-

based education bring an approach and attitude to teaching of focusing on how students

learn and the unique qualities of each learner. Additionally, Chock-Eng and Guild (1998)

deem the overlaps in these theories exist in the following ways: each theory is learner-

centered and promotes diversity; as well teachers and students are reflective practitioners

and decision makers. Therefore, the ideas associated with learning styles, and the

multiple intelligence theory clearly portrays a rationale for brain-based education.

Research documents the need for varied instruction in our classrooms in order to meet the

different ways students' brains learn.

Learning Pure Mathematics with Meaning and Understanding

Teaching pure mathematics with the brain in mind is intended to emphasize

learning mathematics with meaning and understanding. Caine and Caine (1990) support

this idea by accentuating in their writings the objective of brain-based learning is to move

from memorizing information to meaningful learning.

What does it mean to learn pure mathematics with meaning and understanding?

Loewen and Sigurdson (2001) pronounce learning mathematics with meaning is based on

the number of connections students can make between mathematical concepts. When

students learn mathematics with meaning and understanding they trigger previous

networks of knowledge in the brain and make links between these networks and new

mathematical concepts (Loewen and Sigurdson, 2001). Furthermore, Carpenter and

Hiebert (1992) explain the number and the strength of the connections determine the

degree of understanding.

16

Teaching for meaning and understanding can be achieved using alternate

representations of mathematical concepts (Loewen and Sigurdson, 2001). Such other

forms like visual drawings, context stories, hands-on math manipulatives, and real life

problems increase the opportunity to stimulate prior mathematical knowledge networks in

the brain. Varied teaching methods activate different parts of the brain and student's

learning styles that enable students to construct knowledge through the connection of

mathematical concepts to ensure learning with meaning and understanding.

Other educational researchers enlighten us with their ideas about learning with

meaning and understanding in very similar ways. Carpenter and Lehrer (1999) point out

learning math with meaning and understanding is important because when students

acquire knowledge with understanding they can transfer that knowledge to learn new

concepts and solve novel problems. They believe learning with meaning and

understanding is developed through constructing relationships, extending and applying

mathematical knowledge, reflecting about learning experiences, communicating what one

knows, and making mathematical knowledge one's own (Carpenter and Lehrer, 1999).

Checkley (1997) discovers, while interviewing Howard Gardner, learning

mathematics with understanding means students can take ideas they learn in school or

anywhere and apply those appropriately in new situations. We know students understand

something when they can represent the knowledge in more than one way. Kalbfleisch and

Tomlinson (1998) agree and continue to comment how learners should spend more time

constructing their knowledge so they can witness the relationship between the parts and

17

the whole of a concept. Brandt (1994), while conversing with Magdalene Lampert, refers

to understanding as sense making. Learning with understanding and meaning is merely

sense making by traversing the same territory in lots of different directions. Also, Brandt

(1994) reminds us how many more connections are made between mathematical concepts

during daily review, especially if you can do it with other people and talk about different

perspectives.

Teaching pure mathematics with meaning and understanding is an achievable

outcome for brain-based learning. A variety of learning situations and brain stimulations

will enable students to construct knowledge through the connections they make between

previously learned and new mathematical concepts.

Facilitating Learning: Brain-Based Teaching

A brain-compatible mathematics classroom uses what research says about how

students learn to improve teaching. Many of the findings in brain research validate

effective teaching practices. Brandt and Wolfe (1998) suggest that educators, because of

their experience and knowledge base, are in the best position to recognize what brain

research is saying to supplement, explain, or confirm current practices. Brain-based

teaching highlights certain aspects of learning such as connections to previous

knowledge; attention and learning; emotion and learning; movement and learning;

rehearsal and learning; memory and learning; elaboration and learning. These brain-based

ideas are interwoven throughout the strategies teachers' use in the classroom. Brain

strategies match how the brain learns best.

18

Applying Brain Research to Classroom Practice

Researchers do expect teachers to be cautious when interpreting and applying

brain research to their teaching practices. Some researchers more than others warn

teachers about the application of brain research. For instance, Bruer (1998) advises

teachers not to link brain research in any meaningful way to educational practice because

neuroscience does not know enough about brain development and neural function.

Furthermore, Bruer (1997) admits neuroscience has discovered a great deal about the

brain, but argues not nearly enough to guide educational practice. This project contributes

to linking neuroscience research to educational knowledge. On the other hand, Brandt

(1999) responds to Bruer's articles on brain research with his own argument. Brandt

(1999) feels findings from neuroscience are yielding additional insights into the learning

process if used in conjunction with other sources, and educators would be foolish to

ignore this growing body of knowledge. Using neuroscience in the classroom is

challenging because teachers cannot rely exclusively on brain research. Students are

complex and unique; not just one strategy will work. Caine (2000) considers integrating

brain research with other fields of research can provide teachers with a foundation for

excellent teaching.

There is a spectrum of opinions for and against the application of brain research to

educational practice. Nevertheless, most educational researchers portray a common

theme when analyzing the use of brain research in the classroom. The purpose of

neuroscience research is to learn how the brain functions; the purpose is not to tell

teachers what they should be doing in the classroom to increase student understanding.

Brandt (1999) believes if we use the findings about the physical brain with our

19

knowledge about human behaviour and learning we have derived from psychology, social

sciences, along with educational research and professional experience it can further

illuminate our understanding of how students' learn best. As well, Caine and Caine

(1998,2000), and Jensen (1998, 2000) explain there is a vast body of knowledge of brain

research out there but it is up to educators to carefully interpret what brain science means

for classroom practice. Teachers possess an immense background of knowledge about

teaching and learning. Teachers gain this knowledge from educational research, cognitive

science, and classroom experience. Therefore, teachers are in the best position to analyze

and interpret how brain research applies to classroom practice. Brandt (2000) challenges

teachers to keep informed about current brain research so they can provide the best

possible learning environment for their students.

How Our Brains Learn

Teaching strategies will help our brains learn concepts. But, what is actually

happening in our brains when we gain knowledge. Hardiman (2001) explains learning

occurs through the growth of neural connections stimulated by the passage of electrical

current along nerve cells and enhanced by chemicals discharged into the synapse between

neighbouring cells. In more general terms, Wolfe (1999) comments as the brain receives

information from other cells connections are made between the cells. The more

connections you make the more you will remember. Teachers need to strengthen these

neural connections in their students. Wolfe (1999) reminds teachers neurons that fire

together wire together. Thus, learning involves both the attainment of information and the

ability to retrieve the information. The more teachers use various strategies to make

connections in students' brains the more students will remember and understand what

they have learned.

The Importance of Teachers Using Previous Learned Knowledge

20

One of the most challenging things to do as a teacher is to take rather meaningless

information and make it meaningful for our students. In seconds of information entering

our brains, the brain will decide if the information is meaningful. Wolfe (1999) suggests

in order to make meaning out of new information our brains must have previous

information to reactivate connections that were made before. The first time a neuron

makes a connection in our brains a certain amount of energy will be needed (Wolfe,

1999). As further connections are made, less and less energy is needed. Eventually the

connection is automatic and this produces a strong memory. Brandt (2000) agrees with

the idea of using less brain energy when performing familiar functions than when

learning new concepts. This is why the use of prior knowledge is so important when

learning something new. If a teacher is able to link a new concept to something

previously learned less brain energy will be needed to make the new concept a wired

memory. In order for teachers to create meaning in their classrooms every encounter with

something new requires the brain to fit the novel information into an existing connection

of neurons (Wolfe, 1999). Therefore, learning new information must occur within the

context of what the learner already knows.

Everyday our brains make the decision to keep or get rid of information. A major

factor as to whether or not a brain hangs on to new information is if the brain recognizes

a previous pattern. Westwater and Wolfe (2000) declare our brains grasp new data by

searching through previously established neural networks to see whether it can find a

21

place to fit the new information. Educational researchers, Hardiman (2001), Jensen

(1998), King-Friedrichs (2001), Lowery (1998), Sousa (2001), Westwater and Wolfe

(2000), have demonstrated that previous knowledge enhances the understanding of new

information. Westwater and Wolfe (2000) assume if the brain can retrieve stored

information that is similar to new information, the brain is more likely to make sense of

the new information. Furthermore, Sousa (2001) recalls the importance of teachers

helping students to transfer information from what the learner knows to other new

concepts. As well, Lowery (1998) reminds teachers when teaching a new concept to your

students make sure they can assimilate the concept into their prior constructions.

Therefore, in order for our brains to keep new information teachers must be aware of the

significance of previously learned knowledge.

Brain Research: Attention and Learning

What does brain research say about paying attention? Cho and Sylwester (1992-

93) agree teachers should adapt their instructional strategies to meet students' attentional

needs. There are some essential principles to remember when trying to catch students'

attention. Erlauer (2003) contends the first ten minutes of a lesson is when students will

learn the most so you do not want to review during this time. Teachers should hit the

students with the new stuff and then later on in the lesson tie in the review from previous

lessons to relate new concepts to background knowledge. Additionally, Erlauer (2003)

has faith in the fact students need a break in concentration every 20 minutes in order to

keep student attention. Teachers need to shift activities at least every 20 minutes to

maintain the attention of their students. Nevertheless, teachers should still teach the same

concept but in different ways to ensure understanding by all students. Jensen (1998)

22

deems students can only hold their attention in short bursts; no longer than their age in

minutes. Brain-based classrooms usually include fifteen minutes of teacher time used for

key ideas, directions, lectures, reviews, or closings, and the rest of the time is used for

student processing of the new information in their brains through various learning

activities such as projects, discussions, group work, writing, etc (Jensen, 1998).

Furthermore, D' Arcangelo (2000) feels teachers can use emotion to direct

attention, and that attention will lead to better learning. Teachers must use imaginative

teaching strategies like games and competitions to increase students' attention on

unemotional learning activities. For example, teachers should use fun energizing rituals

for class openers, closings, and most of the repetitious classroom work (Jensen, 1998).

Thus, repetitive sedentary activity is followed by an active, enjoyable activity. These

strategies will help grab students' attention when teaching the abstract concepts in pure

math.

Brain Research: Emotion and Learning

Learning is strongly influenced by emotion. Sousa (2001) states emotions are an

integral part of learning, retention, and recall. The stronger an emotion connected to an

experience the stronger the memory of that experience (Brandt and Wolfe, 1998).

Furthermore, Jensen (2000) reveals powerful evidence that embedding intense emotion

such as those that occur with celebrations, competitions, drama, in an activity may

stimulate the release of adrenaline, which may more strongly encode the memory of the

learning. But, if the emotion is too strong, then learning is decreased. Caine and Caine

(1998) cite an important aspect in teaching for meaning includes a state of mind in the

learner characterized by low threat and high challenge. When threat and fear are

23

relatively absent, the brain appears to be able to engage in more complex processes.

Emotionally stressful classroom environments are counterproductive because they can

reduce students' ability to learn. In the classroom, stress might be released through

drama, peer support, games, exercise, discussions and celebrations (Jensen, 1998). Good

teachers consider the emotional climate in a classroom to be critical. Teachers should

invest the first few minutes of every class to get the learner into a positive learning state.

Brain Research: Movement and Learning

Sousa's (2001) research told us that 46% of our students are visually preferred

learners, 19% are auditorily preferred learners, and 35% are kinaesthetically preferred

learners. Since usually one third of our classes are kinaesthetic learners, teachers need to

make sure movement and learning are parallel processes in their classrooms. Ronis

(1999) defines bodily-kinaesthetic learner as the ability to use one's whole body to

express ideas and feelings, and capacity to use one's hands to produce or transform

things. Brain Research confirms that physical activity, moving, stretching, walking, can

actually enhance the learning process (Jensen, 2000).

Teachers should encourage their students to move more to learn more. We think

well on our feet because more blood and oxygen go to our brains when we are standing

and moving (Erlauer and Myrah, 1999). Movement increases heart rate and circulation,

which often increases performance. Certain kinds of movements can stimulate the release

of the body's natural motivators: adrenaline, and dopamine. Teachers should help

students learn with movement. For instance, exercise while you are learning is one type

of movement, while others include active lessons, stretching, drama, simulations, games,

etc. As well, Erlauer (2003) comments on how changing locations can have an improved

24

effect on student memory so it provides more vivid memory triggers. Active learning has

significant advantages over sedentary learning. The advantages include learning in a way

that is longer lasting, better remembered, more fun, and that reaches more kind of

learners (Jensen, 2000).

Brain Research: Rehearsal and Learning

Sousa (2001) exclaims rehearsal is essential for retention and learning. Replicated

studies have demonstrated that cells that fire together wire together. King-Friedrichs

(2001) claims neuronal circuits that are continually activated together become stronger

and they require less energy to activate as remembering becomes more automatic.

Teachers must build into the learning context rehearsal activities that will allow students

to recall concepts when needed. Sometimes, the terms rehearsal and practice are used as

synonyms. Lowery (1998) argues practice and rehearsal do not mean the same thing.

Practice is an activity that does the same thing over and over to improve a performance

(Lowery, 1998). Rehearsal is an activity that takes place when people do something again

in a similar but not identical way to reinforce what they have learned while adding

something new (Lowery, 1998). Rehearsal activities are usually simulations where

students draw the concept, write the concept, act out the concept, and solve a problem

with the concept. Using rehearsal as a learning strategy increases the likelihood the

knowledge will be transferable and useful in a variety of ways. Rehearsal activities are

vital to teaching pure mathematics with meaning and understanding. Students must apply

and transfer concepts learned in one unit to other units.

25

Brain Research: Memory and Learning

Memory and learning must be parallel processes to ensure learning with meaning

and understanding. A new concept is usually related to prior knowledge or experience to

be understood. The information is then practiced or manipulated, and used and applied

numerous times before it becomes ingrained in the brain's long-term memory (Erluaer,

2003). Student's memories are not stored as one memory; students must reconstruct the

memory. Wolfe (1999) indicates the human brain can only work with seven (plus or

minus two) bits of information at one time. Therefore, student's recollections are mostly

at the mercy of their elaborative rehearsal strategies. These strategies encourage learners

to elaborate on the information being learned in a manner that increases understanding

and retention of that information. Brandt (1999) contends only those aspects of

experience that are targets of elaborative encoding processes have a high likelihood of

being remembered.

Both rehearsal and reflection time is essential for long-term memory. Students

need conceptual understanding of how a new skill works. During rehearsal sessions,

students should be reflecting on the new concept and adapting or reshaping it. This

deeper examination of the new concept, as well as the distinct ways of rehearsing through

reshaping will promote useful application of that knowledge in the future (Erluaer, 2003).

Through these processes, learning pure mathematics with meaning and understanding can

be achieved.

Brain Research: Elaboration and Learning

The brain-based approach to learning incorporates elaborative rehearsal strategies.

Wolfe (2001) describes how these strategies encourage learners to elaborate on the

26

information being learned in a manner that increases understanding and retention of that

information. These strategies make the information more meaningful and relevant to the

learner. Jensen (1998) contends teachers need elaboration activities to strengthen their

original contact with students because a synaptic connection is often temporary

Additionally, King-Friedrichs (2001) explains students' recollections are largely at the

mercy of their elaborations. Learning experiences should be elaborate and as deep as

possible using a variety of learning experiences to help students deepen their

understanding.

Brain Research: Elaborative Rehearsal Strategies

The possibilities for elaborating the curriculum and making learning meaningful

to students are endless. These examples are but a few of the many brain-congruent

activities available to teachers. If the content is rigorous and relevant, debates,

storytelling, art, music with rhymes and rhythms, writing and illustrating books,

constructing models, video making, visuals, active review, simulations, projects, hands-

on activities, interactive note taking, drama, games, mnemonics, and graphic organizers

can dramatically enhance student understanding (Jensen, 1997). Furthermore, chunking is

an elaborative rehearsal strategy that uses association so more than seven bits of

information can be remembered at one time. Westwater and Wolfe (2000) deem

understanding that we create new neural networks through experience gives us a second

avenue for making the curriculum meaningful. Since our strongest neural networks are

formed from actual experience, we should involve students in solving authentic problems

in their school or community. As well, analogies, metaphors, and similes are excellent

ways to help the brain find links between new data and previously stored knowledge.

27

Journal writing to help students explore what they do or do not understand, and help them

apply new concepts to their own experience. Mett (1989) discusses how writing in

mathematics can be used at an intermediate time to make a transition, allow absorption of

a new idea, or to personalize a theory by creating examples and applications. All of these

brain-based learning and teaching strategies can be used to teach pure mathematics with

meaning and understanding.

Brain-Based Teaching and Learning Models

The following depictions are an overview of brain-based teaching and learning

models present in educational research. Although there are some differences in their

work, many of their thoughts about brain-based teaching and learning models are similar.

Researchers: Caine and Caine (1990,1995,1997,1998)

Educators who become aware of recent research on how the brain learns will gain

exciting ideas about conditions and environments that can optimize learning. Caine and

Caine (1990, 1995, 1997. 1998) offer the following brain principles as a general

theoretical foundation for brain-based learning. These principles are simple and

neurologically sound. Applied to education, they help us to reconceptualize teaching by

taking us out of traditional frames of reference and guiding us in defining and selecting

appropriate programs and methodologies. All conclusions reached are derived from and

rest on the work of many researchers and research fields, including neurosciences,

cognitive science, psychology, social sciences, and education. The brain principles are as

follows: the brain is a complex adaptive system: body, mind and brain are one dynamic

unity; the brain is a social brain; the search for meaning is innate; the search for meaning

occurs through patterning; emotions are critical to patterning; the brain/mind processes

28

parts and wholes simultaneously; learning involves both focused attention and peripheral

perception; learning always involves conscious and unconscious processes; we have at

least two ways of organizing memory: a spatial memory system and a set of systems for

rote learning; learning is developmental; complex learning is enhanced by challenge and

inhibited by threat; each brain is uniquely organized. These brain/mind-learning

principles should be implemented into a climate of: relaxed alertness, meaningful

experience, and active processing.

Researcher: Jensen (1998)

Many times we must reteach things because we do not teach in ways that match

how our students' brains learn. Jensen (1998) mentions five brain-based principles to

follow in order to understand the brain's learning process: neural history, context,

acquisition, elaboration, and encoding. Neural history influences how students learn

based on prior learning, character, environment, peers, and life experience. Learning

context involves the emotional climate of the classroom. Unexpressed emotions can

inhibit learning. Therefore, teachers should encourage activities, such as drama,

discussions, and celebrations so students have an outlet for emotional expression in

classrooms. As well, teachers need to recognize the majority of the acquisition of

knowledge by students comes to them indirectly. Students should spend time processing

what they have learned through projects, discussions, group work, partner work, writing,

design, and feedback. Direct teaching should be limited to fifteen minutes per class.

Jensen (1998) exclaims students who do the talking and the doing do the learning.

Elaboration activities are needed so students can strengthen their neural connections with

the new material being learned for deeper understanding. Finally, teachers need to

permanently encode a classes learning in students' brains.

Researcher: Erlauer (2003)

29

Erlauer (2003) developed brain-based principles from what she knows about

learning in order to improve teaching. The brain-based principles include the importance

of the following in a classroom: emotion, movement, relevant content, interesting

instructional strategies, value of student choice, time, and collaboration. Emotion deals

with how students use emotions to remember their learning. Movement helps students

learn in several ways. Even changing locations can have an improved effect on student

memory as it provides more vivid memory triggers. Relevant content is significant to the

learning process because the brain remembers information that is meaningful and linked

to prior knowledge or experience. Teachers are challenged to make what students are

learning interesting and be able to demonstrate how that information is relevant to them.

Providing interesting instructional strategies is important to the learning process, as it is a

method to gain student attention, and to meet different student learning styles. The value

of student choice on learning activities allows students to work with different multiple

intelligences to ensure understanding for all students. Time is essential for rehearsal and

reflection activities for long-term memory. Lastly, collaboration in learning is a key

teaching strategy as the brain learns socially.

Researcher: Hardiman (2001)

Hardiman (2001) links how the mind works during learning and current brain

research to produce best practices for teaching all students. Also, Hardiman (2001)

explains how the use of this model has resulted in exciting learning experiences for

30

students as well as increased scores on state assessments. The model includes acquiring

and integrating knowledge, extending and refining knowledge, using knowledge

meaningfully, and habits of mind. Learning requires both the acquisition of information

and the ability to retrieve and reconstruct that information whenever necessary.

Extending and refining knowledge requires the brain use multiple and complex systems

of retrieval and integration. Using knowledge meaningfully includes activities that

require students to make decisions, investigate, conduct experiments, and solve real

world problems. To end with, habits of mind involve mental habits that enable students to

facilitate their own learning. For example, monitoring one's own thinking, goal setting,

maintaining one's own standards of evaluation, self-regulating, and applying one's

unique learning style to future learning situations.

Researcher: King-Friedrichs (2001)

King-Friedrichs (2001) introduces brain-based techniques for improving memory

and learning. The brain-based techniques include: using emotion, connecting to prior

knowledge, making sense, elaborating on key concepts, and incorporating rehearsal

activities. When students are emotionally engaged with learning, certain

neurotransmitters in the brain signal to the hippocampus, a vital brain structure involved

with memory, to stamp this event with extra vividness. Teachers should connect new

learning to prior knowledge because it is easier for the brain to reactivate neural

connections than it is to learn something with no previous understanding. Students should

make sense of new concepts by using writing to challenge students to organize and

articulate what they have just learned. Teachers need to construct activities that allow

students to elaborate on key concepts using a variety of learning experiences to help

31

students deepen their understanding of the new material. Rehearsal activities are

important to learning. Neural connections that are continually activated together become

stronger and require less energy to activate as remembering becomes more automatic.

Teachers must use rehearsal activities that will likely be present when students need to

remember the new material.

Chapter Summary

The chapter examined the ideas, thoughts, and beliefs of many educational

researchers on topics connected to brain-based teaching and learning. Students' learning

styles and the multiple intelligence theory layout the rationale for brain-based teaching

and learning. One of the outcomes of the application of brain research to classroom

practice is learning pure mathematics with meaning and understanding. A brain-

compatible mathematics classroom uses what research says about how students learn to

improve teaching. Nevertheless, researchers do expect teachers to be cautious when

interpreting and applying brain research to their teaching practices. Educational

researchers recommend teachers should be aware of certain aspects of brain-based

teaching and learning such as being cognitive of how a student's brain learns; connecting

previous learned knowledge to new concepts; gaining attention; using emotion and

movement to drive learning; and stressing rehearsal, memory, and elaboration activities

to motivate learning.

Chapter 3: Brain-Based Teacher Resource

"Students are resourceful problem solvers working alone or in groups productively and

reflectively, with the skilled guidance of their teachers. Students communicate their ideas

orally and in writing. Students value mathematics and engage actively in learning it."

(National Council of Teachers of Mathematics, 2000, p. 3)

Introduction

This chapter applies brain-based learning and teaching strategies to classroom

practice in order to produce a practical brain-based resource for pure mathematics

teachers. The following section demonstrates how to teach pure mathematics with the

brain in mind. The resource includes the algebraic unit of study that investigates

quadratic functions and equations. The reason this unit was chosen is because a lot of the

mathematical concepts in this section require abstract thinking about theoretical ideas.

Therefore, many students find this part of the mathematics program to be challenging and

frustrating. The resource uses brain-based teaching, learning, and elaborative rehearsal

strategies to teach these algebraic mathematical concepts that enable students to learn

mathematics with meaning and understanding.

The Underlying Learning Principles for the Resource

Recognize Learning Styles

The uniqueness present in all human beings is reason enough why one approach

to classroom teaching is ineffective for most students. Classrooms need to be responsive

to students' varying readiness levels, varying interests, and varying learning profiles.

Teachers should feel a responsibility to present information in varied ways, for example,

orally, visually, through demonstration, movement, and hands-on activities if they want

32

to give all their students an opportunity to learn. Therefore, teachers can stimulate and

facilitate learning in all children by addressing the need to know in multiple ways. By

providing alternate ways of learning, students are at a liberty to gain new information

through those learning styles most comfortable for them.

Learning Pure Mathematics with Meaning and Understanding

33

Teaching pure mathematics with the brain in mind is intended to emphasize

learning mathematics with meaning and understanding. Learning mathematics with

meaning is based on the number of connections students can make between mathematical

concepts. When students learn mathematics with meaning and understanding they trigger

previous networks of knowledge in the brain and make links between these networks and

new mathematical concepts. Teaching for meaning and understanding can be achieved

using alternate representations of mathematical concepts. Such other forms like visual

drawings, context stories, hands-on math manipulatives, and real life problems increase

the opportunity to stimulate prior mathematical knowledge networks in the brain. Varied

teaching methods activate different parts of the brain and student's learning styles that

enable students to construct knowledge through the connection of mathematical concepts

to ensure learning with meaning and understanding.

Brain-Based Teaching and Learning

A brain-compatible mathematics classroom uses what research says about how

students learn to improve teaching. Many of the findings in brain research validate

effective teaching practices. Brain-based teaching highlights certain aspects of learning

such as connections to previous knowledge, attention, emotion, movement, rehearsal,

34

memory, and elaboration. These brain-based ideas are interwoven throughout the

strategies teachers' use in the classroom. Brain strategies match how the brain learns best.

Framework: Brain-Based Daily Lessons

The following brain-based teaching and learning strategies will form the

framework for each daily math lesson in the resource. These ideas will be implemented

into daily learning activities. Every part of the daily lesson is built around the brain-based

framework. From the attention and goal activities to the rehearsal, and debriefing

activities; all activities reflect brain-based teaching and learning principles.

Brain-Based Strategy #1: Previous Learned Knowledge

The brain uses less energy when performing familiar functions than when

learning new concepts. This is why the use of prior knowledge is so important when

learning something new. If a teacher is able to link a new concept to something

previously learned less brain energy will be needed to make the new concept a wired

memory. In order for teachers to create meaning in their classrooms every encounter with

something new requires the brain to fit the novel information into an existing connection

of neurons. Therefore, learning new information must occur within the context of what

the learner already knows.

Brain-Based Strategy #2: Attention and Learning

There are some essential principles to remember when trying to catch students'

attention. The first ten minutes of a lesson is when students will learn the most so you do

not want to review during this time. Teachers should hit the students with the new stuff

and then later on in the lesson tie in the review from previous lessons to relate new

concepts to background knowledge. Additionally, students need a break in concentration

35

every 20 minutes. Teachers need to shift activities at least every 20 minutes to maintain

the attention of their students. Nevertheless, teachers should still teach the same concept

but in different ways to ensure understanding by all students. Brain-based lessons

consistently use the idea that students can only hold their attention in short bursts; no

longer than their age in minutes. Brain-based classrooms usually include fifteen minutes

of teacher time used for key ideas, directions, lectures, reviews, or closings, and the rest

of the time is used for student processing of the new information in their brains through

various learning activities such as projects, discussions, group work, and writing. As

well, teachers should use imaginative teaching strategies like games and competitions to

increase students' attention on unemotional learning activities. For example, teachers

should use fun energizing rituals for most of the repetitious seatwork in math. These

strategies will help grab students' attention and increase their motivation to do the

important reinforcing concept work.

Brain-Based Strategy #3: Emotion and Learning

Learning is strongly influenced by emotion. The stronger an emotion connected to

an experience the stronger the memory of that experience. Embedding intense emotion

such as those that occur with celebrations, competitions, drama, in an activity may

stimulate the release of adrenaline, which may more strongly encode the memory of the

learning. But, if the emotion is too strong, then learning is decreased. Emotionally

stressful classroom environments are counterproductive because they can reduce

students' ability to learn. In the classroom, stress might be released through drama,

cooperative learning, games, exercise, discussions and celebrations. As well, the learning

activity Think Pair Share is a brain-based strategy that considers the relationship between

36

emotion and learning. Bennett and Rolheiser (2001) believe when students have time to

think and then share with a partner before sharing with the entire class, they are more

likely to feel secure and experience success.

Brain-Based Strategy #4: Movement and Learning

Teachers should encourage their students to move more to learn more. We think

well on our feet because more blood and oxygen go to our brains when we are standing

and moving. Certain kinds of movements can stimulate the release of the body's natural

motivators: adrenaline, and dopamine. Teachers should help students learn with

movement. For instance, exercise while you are learning is one type of movement, while

others include active lessons, stretching, drama, simulations, games, etc. Active learning

has significant advantages over sedentary learning. The advantages include learning in a

way that is longer lasting, better remembered, more fun, and that reaches more kind of

learners.

Brain-Based Strategy #5: Rehearsal and Learning

Rehearsal is essential for retention and learning. Rehearsal is an activity that takes

place when people do something again in a similar but not identical way to reinforce

what they have learned while adding something new. Teachers must build into the

learning context rehearsal activities that will allow students to recall concepts when

needed. Rehearsal activities are usually simulations where students draw the concept,

write the concept, act out the concept, and solve a problem with the concept. Using

rehearsal as a learning strategy increases the likelihood the knowledge will be

transferable and useful in a variety of ways.

37

Brain-Based Strategy #6: Memory and Learning

Memory and learning must be parallel processes to ensure learning with meaning

and understanding. A new concept is usually related to prior knowledge or experience to

be understood. The information is then practiced or manipulated, and used and applied

numerous times before it becomes ingrained in the brain's long-term memory. Student's

recollections are mostly at the mercy of their elaborative rehearsal strategies. These

strategies encourage learners to elaborate on the information being learned in a manner

that increases understanding and retention of that information. Both rehearsal and

reflection time is essential for long-term memory. Students need conceptual

understanding of how a new skill works. During rehearsal sessions, students should be

reflecting on the new concept and adapting or reshaping it. This deeper examination of

the new concept, as well as the distinct ways of rehearsing through reshaping will

promote useful application of that knowledge in the future.

Brain-Based Strategy #7: Elaboration and Learning

The brain-based approach to learning incorporates elaborative rehearsal strategies.

These strategies encourage learners to elaborate on the information being learned in a

manner that increases understanding and retention of that information. These strategies

make the information more meaningful and relevant to the learner. Learning experiences

should be elaborate and as deep as possible using a variety of learning experiences to

help students deepen their understanding. Some elaborative rehearsal strategies are

debates, storytelling, art, music with rhymes and rhythms, writing and illustrating books,

constructing models, video making, visuals, active review, simulations, projects, hands-

on activities, interactive note taking, drama, games, mnemonics, and graphic organizers

38

can dramatically enhance student understanding. Furthermore, chunking is an elaborative

rehearsal strategy that uses association so more than seven bits of information can be

remembered at one time. Since our strongest neural networks are formed from actual

experience, we should involve students in solving authentic problems in their school or

community. As well, analogies, metaphors, and similes are excellent ways to help the

brain find links between new data and previously stored knowledge. Journal writing to

help students explore what they do or do not understand, and help them apply new

concepts to their own experience.

Brain-Based Strategy #8: Collaboration and Learning

The brain is a social brain. Intelligence is enhanced by social situations. The brain

needs to be in situations where it is allowed to experience talk, teamwork, debate, and

opinions. Cooperative learning activities support collaborative encounters. Therefore,

activities like Think Pair Share where students are given time to think and then move on

to sharing their ideas with someone else uses our social brain to develop a deeper

understanding of what they are learning.

Brain-Based Lesson Plans

Each lesson in the teacher resource incorporates these brain strategies so students

are able to learn pure mathematics with meaning and understanding. A pure mathematics

20 quadratic equation and function project is presented next. All lessons have a project

work section. Students will work on this project daily throughout the unit. The daily

project work enables students to apply their mathematical concepts to a real life situation.

Daily brain-based lessons follow the project.

39

Mathematics Project: Quadratic Equations and Functions

Pure Mathematics 20 Quadratic Equation and Function Project

(Addison-Wesley, 2000, p. 42-47)

The Task:

In this project, you will design a golf video game. Your group will include three

students. You will work with graphs of quadratic functions, characteristics of quadratic

functions, and equations of quadratic functions. You will use your graphing calculator to

find the maximum and x-intercepts of the graph of a quadratic function.

Background:

When designers and programmers develop video games they must make

movement realistic. Whether it is in the flight of an object, the action of a ski jumper, or

the movement of a golf ball, players of the game expect realism.

Many video games involve projectiles such as footballs, golf balls, or ski jumpers.

A projectile is any object that is thrown, shot or propelled in some way into the air. Most

projectiles are propelled at an angle, and move horizontally as well as vertically. The

curved path, or trajectory, of a projectile is affected by the initial speed and inclination of

a shot or kick, by gravity, and by air resistance. Gravity pulls everything on Earth toward

the centre of Earth. Air resistance slows the projectile. Video game programmers use

40

formulas that have been determined by engineers to model the flights of projectiles so their games seem real.

Getting Started:

When a golfer hits the golf ball squarely at the bottom of the swing, with the shaft

perpendicular to the ground, the ball is propelled into the air in a direction perpendicular

to the face of the club. For a 7 iron, the clubface is set at an angle of 38 with respect to

the shaft of the club. The initial speed of the ball can be determined from stroboscopic

photographs.

The speed of an object moving at an angle to the vertical can be resolved into its

horizontal and vertical components, provided we know the angle of travel. Horizontal and

vertical speeds are determined as the sides of a right triangle by the use of trigonometric

ratios. 37m!s

29.2m!s

An engineer used photographs of a professional golfer to determine the average

initial speeds for a variety of clubs. She then calculated the corresponding vertical and

horizontal speeds. Her results are shown in the table below.

Club Club Angle Initial Speed Horizontal Vertical Speed Mis Speed rnJs rnJs

Driver 9.5 86 84.8 14.2

3 wood 12 71 69.4 14.8

3 iron 20 51 47.9 17.4

5 iron 29 42 36.7 20.4

7 iron 38 37 29.2 22.8

9 iron 47 34 23.2 24.9

Pitching Wedge 56 32 17.9 26.5

Sand Wedge 65 31 13.1 28.1

Quadratic Equation:

She found that, for the 7iron, the equation of the trajectory of the ball could be

approximated by this quadratic equation:

y = -4.9(X/29.2)2 + 22.8 (X/29.2)

y = -4.9(xlHor. Speed)2 + Ver. Speed (xlHor. Speed)

Hor. is short for Horizontal

V er. is short for Vertical

Where y metres is the height, and x metres is the horizontal

distance down the fairway.

Student Questions:

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1. The constant - 4.9 in the equation is the engineer's way of accounting for the effect of

gravity. Why is the number negative?

2. Where does the horizontal speed of the 7 iron appear in the formula? Where is the

vertical speed?

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3. Use your graphing calculator to graph the trajectory of a golf ball hit into the air by a 7

iron.

4. Use the CALC feature of the graphing calculator to determine the maximum height of

the ball, to the nearest metre.

5. Determine the maximum horizontal distance the ball travels while in the air. Why may

this not be the total horizontal distance travelled by the ball?

6. Change the constants in the equation to those appropriate for a driver. Graph the

trajectory this model predicts for a golf ball struck by a driver. Record the maximum

height and distances.

7. Consider the layout of the par-4 golf hole on a golf course shown below. Assume a

two-putt for the hole. One possible line of approach is shown. Calculate the distance for

each shot. Would this line of approach be suitable for the club selection: driver, 7 iron,

putter? Explain. Suggest other possible approaches.

Water Par 4 hole

Scale I I I Om 20m

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Project Requirements and Presentation:

To design video games successfully, the programmers and course designers must

work together to produce a course design that is interesting and fun to playas a video

game. The programmers must use formulas that imitate the movement of a golf ball hit

by different clubs and at different speeds. The course designers must make a course that

is interesting to play, given the clubs available and how they are hit. Courses that are too

easy are boring and those that are too hard aren't much fun either.

1. Make a table containing the flight data for each club. The flight data includes how

far and how high the ball will travel under the conditions described in the table for

each club. Using the quadratic equation given, graph the quadratic function for

each club on your graphing calculator. Find how far and how high the ball will

travel for each club using the zero and maximum functions on the graphing

calculator.

2. Design three holes of a golf course that would be suitable for inclusion in a video

game. Use